All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$2.50$ each for students, and the group paid $$37.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$17.00$ each for teachers and $$9.00$ each for students, and the group paid $$106.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2.5y = 37}$ ${17x+9y = 106}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-17x-5y = -74}$ ${17x+9y = 106}$ Add the top and bottom equations together. $ 4y = 32 $ $ y = \dfrac{32}{4}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {8.5x+2.5y = 37}$ to find $x$ ${8.5x + 2.5}{(8)}{= 37}$ $8.5x+20 = 37$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 8}$ into $ {17x+9y = 106}$ and get the same answer for $x$ ${17x + 9}{(8)}{= 106}$ ${x = 2}$ There were $2$ teachers and $8$ students on the field trips.